A328725 -id:A328725 - cp's OEIS frontend

A005260 a(n) = Sum_{k = 0..n} binomial(n,k)^4.

1, 2, 18, 164, 1810, 21252, 263844, 3395016, 44916498, 607041380, 8345319268, 116335834056, 1640651321764, 23365271704712, 335556407724360, 4854133484555664, 70666388112940818, 1034529673001901732, 15220552520052960516, 224929755893153896200, 3337324864503769353060

Offset: 0

N. J. A. Sloane

This sequence is s_10 in Cooper's paper. - Jason Kimberley, Nov 25 2012
Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y*z + x*y*z + w*x + y*z)). - Gheorghe Coserea, Jul 13 2016
This is one of the Apéry-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017
Two walkers, A and B, stand on the South-West and North-East corners of an n X n grid, respectively. A walks by either North or East steps while B walks by either South or West steps. Sequence values a(n) < binomial(2*n,n)^2 count the simultaneous walks where A and B meet after exactly n steps and change places after 2*n steps. - Bradley Klee, Apr 01 2019
a(n) is the constant term in the expansion of ((1 + x) * (1 + y) * (1 + z) + (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n. - Seiichi Manyama, Oct 27 2019

			G.f. = 1 + 2*x + 18*x^2 + 164*x^3 + 1810*x^4 + 21252*x^5 + 263844*x^6 + ...

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 79.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..834 (terms 0..250 from Jason Kimberley)
B. Adamczewski, J. P. Bell, and E. Delaygue, Algebraic independence of G-functions and congruences "a la Lucas", arXiv preprint arXiv:1603.04187 [math.NT], 2016.
Hacene Belbachir and Yassine Otmani, A Strehl Version of Fourth Franel Sequence, arXiv:2012.02563 [math.CO], 2020.
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
W. Y. C. Chen, Q.-H. Hou, and Y-P. Mu, A telescoping method for double summations, J. Comp. Appl. Math. 196 (2006) 553-566, eq (5.5).
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
M. Coster, Email, Nov 1990
E. Delaygue, Arithmetic properties of Apéry-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s10 p. 3.
Darij Grinberg, Introduction to Modern Algebra (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Robert Osburn, Armin Straub, and Wadim Zudilin, A modular supercongruence for 6F5: an Apéry-like story, arXiv:1701.04098 [math.NT], 2017.
M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
Zhi-Wei Sun, Open conjectures on congruences, Nanjing Univ. J. Math. Biquarterly 36(2019), no.1, 1-99. (Cf. Conjectures 49-51.)
Eric Weisstein's World of Mathematics, Binomial Sums
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, preprint of Combinatorics, Probability and Computing, 24(1), 2015, 354-372.
Jin Yuan, Zhi-Juan Lu, and Asmus L. Schmidt, On recurrences for sums of powers of binomial coefficients, J. Numb. Theory 128 (2008) 2784-2794.

Column k=4 of A309010.
Cf. A000172, A096192, A328725.
Related to diagonal of rational functions: A268545-A268555.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Row sums of A202750.

A005260 := proc(n)
        add( (binomial(n,k))^4,k=0..n) ;
end proc:
seq(A005260(n),n=0..10) ; # R. J. Mathar, Nov 19 2012

Table[Sum[Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 09 2014 *)

{a(n) = sum(k=0, n, binomial(n, k)^4)};

def A005260(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m = m*(n-k)**4//(k+1)**4
    return g # Chai Wah Wu, Oct 04 2022

a(n) ~ 2^(1/2)*Pi^(-3/2)*n^(-3/2)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
D-finite with recurrence: n^3*a(n) = 2*(2*n - 1)*(3*n^2 - 3*n + 1)*a(n-1) + (4*n - 3)*(4*n - 4)*(4*n - 5)*a(n-2). [Yuan]
G.f.: 5*hypergeom([1/8, 3/8],[1], (4/5)*((1-16*x)^(1/2)+(1+4*x)^(1/2))*(-(1-16*x)^(1/2)+(1+4*x)^(1/2))^5/(2*(1-16*x)^(1/2)+3*(1+4*x)^(1/2))^4)^2/(2*(1-16*x)^(1/2)+3*(1+4*x)^(1/2)). - Mark van Hoeij, Oct 29 2011
1/Pi = sqrt(15)/18 * Sum_{n >= 0} a(n)*(4*n + 1)/36^n (Cooper, equation (5)) = sqrt(15)/18 * Sum_{n >= 0} a(n)*A016813(n)/A009980(n). - Jason Kimberley, Nov 26 2012
0 = (-x^2 + 12*x^3 + 64*x^4)*y''' + (-3*x + 54*x^2 + 384*x^3)*y'' + (-1 + 40*x + 444*x^2)*y' + (2 + 60*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016
For r a nonnegative integer, Sum_{k = r..n} C(k,r)^4*C(n,k)^4 = C(n,r)^4*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016
a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], 1). - Peter Luschny, Jul 27 2016
Sum_{n>=0} a(n) * x^n / (n!)^4 = (Sum_{n>=0} x^n / (n!)^4)^2. - Ilya Gutkovskiy, Jul 17 2020
a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2k,k)*C(2n-2k,n-k)*(-1)^(n-k). This can be proved via the Zeilberger algorithm. - Zhi-Wei Sun, Aug 23 2020
a(n) = (-1)^n*binomial(2*n, n)*hypergeom([1/2, -n, -n, n + 1], [1, 1, 1/2 - n], 1). - Peter Luschny, Aug 24 2020
a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,n)*binomial(2*n-k,n) [Theorem 1 in Belbachir and Otmani]. - Michel Marcus, Dec 06 2020
a(n) = [x^n] (1 - x)^(2*n) P(n,(1 + x)/(1 - x))^2, where P(n,x) denotes the n-th Legendre polynomial. See Gould, p. 66. This formula is equivalent to the binomial sum identity of Zhi-Wei Sun given above. - Peter Bala, Mar 24 2022
From Peter Bala, Oct 31 2024: (Start)
For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^3 * binomial(n-1, k).
For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n, -n + 1], [1, 1, 1], 1). (End)
G.f.: Sum_{k>=0} Sum_{l=0..p*k} Sum_{m=0..l} (-1)^m*binomial(p*k+1,m)*binomial(l+k-m,k)^p*x^(l+k)/(1-x)^(p*k+1), where p = 4. - Miles Wilson, Apr 12 2025

Edited by Michael Somos, Aug 09 2002

Minor edits by Vaclav Kotesovec, Aug 28 2014

A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 841, 4201, 25705, 118441, 423721, 1628881, 8065201, 41225185, 184416961, 768211081, 3420474121, 16620237001, 79922011465, 364149052705, 1638806098945, 7655390077105, 36739991161105, 174363209490625, 811840219629121, 3790118889635521

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (x^4 + y^4 + z^4 + w^4 + x*y*z*w)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1429

Cf. A002426, A344560, A361703.
Cf. A008977, A328725.
Cf. A082488, A274783, A274785.

a(n) = n!*sum(k=0, n\4, 1/(k!^4*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} 1/(k!^4 * (n-4*k)!).
G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(4*k)/(1-x)^(4*k+1).
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) - (n-1)*(6*n^2 - 12*n + 7)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 255*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 5^(n + 3/2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). (End)

A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

Original entry on oeis.org

1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Inverse binomial transform of the Franel numbers (A000172). - Paul D. Hanna, Feb 26 2012
a(n) is the constant term in the expansion of (1 + x + y + 1/x + 1/y + x/y + y/x)^n. - Seiichi Manyama, Oct 26 2019
a(n) is the constant term in the expansion of (-1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
a(n) is the number of n step closed walks on the hexagonal lattice with loops at each node. A step along a loop leaves the position unchanged. The bijection is as follows: after subtracting 1 from each element in the array, values are -1, 0 or 1 and row and column sums are zero. There are only seven possibilities for each row. An all zero row corresponds with a step along the loop leaving the position unchanged and the others to a unit step in each of the six possible directions. This justifies that this sequence is the binomial transform of A002898. - Andrew Howroyd, May 09 2020

			G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...

Andrew Howroyd, Table of n, a(n) for n = 0..500 (terms 1..99 from R. H. Hardin)

Column k=3 of A328747 and A334549.

Cf. A000172, A002898, A006480, A328725.

Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x + x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Feb 26 2012

a(n)={sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3))} \\ Andrew Howroyd, May 09 2020

From Paul D. Hanna, Feb 26 2012: (Start)
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).
Equals the binomial transform of A002898.
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.
(End)
Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012
G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013

a(0)=1 prepended by Andrew Howroyd, May 09 2020

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 7, 1, 0, 1, 1, 15, 31, 19, 1, 0, 1, 1, 31, 115, 175, 51, 1, 0, 1, 1, 63, 391, 1255, 991, 141, 1, 0, 1, 1, 127, 1267, 8071, 13671, 5881, 393, 1, 0, 1, 1, 255, 3991, 49399, 161671, 160461, 35617, 1107, 1, 0

Offset: 0

Seiichi Manyama, Oct 27 2019

T(n,k) is the constant term in the expansion of (-1 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

For fixed k > 0, T(n,k) ~ (2^k - 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

			Square array begins:
   1, 1,  1,   1,     1,      1, ...
   1, 1,  1,   1,     1,      1, ...
   0, 1,  3,   7,    15,     31, ...
   0, 1,  7,  31,   115,    391, ...
   0, 1, 19, 175,  1255,   8071, ...
   0, 1, 51, 991, 13671, 161671, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.
Main diagonal gives A328811.
T(n,n+1) gives A328813.
Cf. A309010, A328748, A328807.

T[n_, k_] := Sum[(-1)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

Original entry on oeis.org

1, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424, 187477238295444829732

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

Column k=4 of A328748.
Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A126869 (m=2), A002898 (m=3), this sequence (m=4), A328751 (m=5).
Cf. A002899, A005260, A328725.

Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^4, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)

{a(n) = polcoef(polcoef(polcoef((-2+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = 2*(n-1)*n*(2*n - 1)*a(n-1) + 112*(n-1)^3*a(n-2) + 184*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 336*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n-4) * 7^(n + 3/2) / (Pi^(3/2) * n^(3/2)). (End)

A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

Original entry on oeis.org

1, 3, 23, 225, 2583, 32133, 422069, 5757699, 80790775, 1158593589, 16905540753, 250185539079, 3746205581589, 56652844671855, 864032059578879, 13274539401672345, 205252378269637815, 3191578469685269925, 49876569284504593505, 782943268394316187815

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Column k=4 of A328807.

Cf. A005260, A328725, A328735.

Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^4, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = polcoef(polcoef(polcoef((1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

{a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)*a(n-1) + (n-1)*(22*n^2 - 44*n + 13)*a(n-2) - 44*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 51*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ sqrt(2) * 17^(n + 3/2) / (64 * Pi^(3/2) * n^(3/2)). (End)

cp's OEIS Frontend

A005260 a(n) = Sum_{k = 0..n} binomial(n,k)^4.

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A361637 Constant term in the expansion of (1 + x + y + z + 1/(x*y*z))^n.

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A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

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A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

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A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

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A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

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A361637 Constant term in the expansion of (1 + x + y + z + 1/(xyz))^n.

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

A328808 Constant term in the expansion of (3 + x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.